What Michael Hopkins thinks about from day to day is algebraic topology, a field that solves topological, or geometrical, problems by translating them into algebra. (Staff photo Kris Snibbe/Harvard News Office)

Michael Hopkins, algebraic topologist

Professor studies the relationships between numbers and shapes

By Bob Brustman
Harvard News Office

When he was a teenager, Michael Hopkins, professor of mathematics, was torn between being a rock star and being a mathematician. "I played guitar and was in a band," he said. "We played rock 'n' roll - we were pretty into Jethro Tull and that kind of stuff. But we weren't that good. I did better in math, in the end."

And he has done well in math: he now holds the reputation as being the world's pre-eminent algebraic topologist. Asked how he felt about this reputation, he shrugged and said, "It's not something I think about from day to day."

What he does think about from day to day is algebraic topology, a field that solves topological, or geometrical, problems by translating them into algebra. "Algebra has to do with numbers and equations," Hopkins said, "and topology means it has to do with shapes of things. Algebraic topology is about a relationship between numbers and shapes."

To demonstrate such a relationship, Hopkins picked up a soccer ball. Soccer balls are covered by hexagons and pentagons - a mathematical shape known as a truncated icosahedron. If you count the number of these shapes, subtract the number of straight lines, and add the number of points where the shapes intersect, you arrive at the number two. "And it's always two," he said, no matter what shapes one chooses to cover the ball with. "That number two is something algebraic that's attached to the sphere," he said. And it doesn't even matter if the sphere remains round: "If the ball was flat, or stretched out, the number is still two - it tells you something about the shape of the ball but it doesn't have to do with its actual roundness." The number two is an "invariant" of the ball.

In this way, algebraic topology attempts to differentiate between multidimensional shapes, or topological spaces. Topological spaces are structures that can be continuously changed; for instance, the ball can be flattened or stretched. "This means that [algebraic topology] is the right language for describing qualitative features of geometry," said Hopkins.

This example with the soccer ball provides what Hopkins calls a "mathematical experience." Algebraic topology is derived from mathematical experiences in much the same way that geometry derives from our physical experience of shapes in the world and algebra from manipulating things.

Hopkins was drawn to algebraic topology as an undergraduate mathematics student because "I liked the level of abstraction involved and because it's very elegant," he said.

This year, Hopkins has been teaching two graduate courses, one on string topology, which involves string theory and algebraic topology, and another course on algebraic topology literature. "It's a kind of literature course for graduate students to learn about learning about algebraic topology," he said. "There are a lot of student presentations and discussion about reading papers and how to get ideas out of papers to work on, how to find new problems to work on. It's a course designed to help people make the transition from being a student to doing their own research."

Hopkins, who was at the Massachusetts Institute of Technology (MIT) for 16 years before joining the Harvard faculty last fall, has been consistently impressed by his students: "Students at both places are awfully good. I don't have the challenges of trying to teach someone who really can't get it; it's more trying to push someone who's really very talented."

He says that he's "blown away" by the students at Harvard and their breadth of knowledge.

Hopkins' Harvard recruitment may even be said to have been begun by students. "While I was at MIT, I would get contacted by Harvard students who were interested in algebraic topology," he said. "The opportunity to come here and work directly with those students and to build up algebraic topology at Harvard was really attractive to me."

"I like watching students grow and evolve," Hopkins said. "I help usher them into the community of mathematicians ... and I really enjoy watching that community grow."

Hopkins joined the community of mathematicians himself when he was an undergraduate at Northwestern University. "There were some good people in algebraic topology there and I really felt that was where I belonged," he said. After receiving his B.A. in mathematics in 1979, he went to Oxford to study and earned a D.Phil. in mathematics in 1984. He later returned to Northwestern, earning his Ph.D. in mathematics in 1984. He joined Princeton University as a postdoctoral fellow, instructor, and assistant professor from 1984 to 1987 and the University of Chicago as a professor from 1988 to 1989. He joined MIT as an associate professor in 1989 and was named a full professor in 1990.

What aspect of his work does he enjoy most? "The thing I like most is that I get to spend as much time as I want thinking about the things that most interest me, which is a fantastic freedom and a fantastic opportunity."